Given a quadratic form $q$, we know that it can be brought into a diagonal form $q=\lambda_1x_1^2+\cdots+\lambda_nx_n^2$. Now we can set $z_i=\sqrt\lambda_i x_i$ and get the form $q=z_1^2+\cdots+z_n^2$. My question is, what process should I follow in order to find the basis corresponding to the last form?
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Are you asking how to find the $x_i$'s? If so, check out this post: http://math.stackexchange.com/questions/1382288/finding-p-such-that-ptap-is-a-diagonal-matrix/1382315#1382315 – Christopher Carl Heckman May 10 '16 at 00:12
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ummm; this would be for a positive form only. I am assuming you are working in the real numbers. – Will Jagy May 10 '16 at 00:38